Some economic experts brought additive scheduling to attending as a really utile tool for economic analysis after the Second World War. Koopmans ( 1951 ) found out the mathematical scheduling job of optimising an nonsubjective map topic to a set of additive inequality restraints. He so defined Pareto-Koopmans proficient efficiency as: ‘a point in the trade good infinite as efficient whenever an addition in the net end product of one good can be achieved merely at the cost of a lessening in the net end product of another good ‘ ( Koopmans, 1951 cited in Ray 2004, p.2 ) . Debreu ( 1951 ) raised the thought of ‘coefficient of resource use ‘ , which he utilised as a step of proficient efficiency for the economic system. In his mathematical theory, due to the inefficient use of resources, the divergence of the step was marked as deadweight loss suffered by the society. Besides, as mentioned before, Farrell ( 1957 ) was the first to use the additive convex attack to frontier appraisal. He constructed additive programming theoretical account with existent informations of input and end product from certain sample house ( bank ) . The mathematical consequence of the theoretical account interpreted the numerical step of the proficient efficiency of the selected sample house or bank. He introduced the thought that allocative efficiency is another of import portion when mensurating overall economic efficiency. Furthermore, he recognized the restriction of utilizing ordinary least square arrested development to gauge production frontier, which was because the consequence was non sole as the ascertained points would be on both sides of the map. The solution he took was ‘taking a nonparametric attack and approximated the implicit in production possibility set by the bulging hull of a cone incorporating the ascertained input-output packages ‘ ( Ray 2004, p.3 ) .

Subsequently on, there were several other economic experts proposed more methods: Aigner and Chu ( 1968 ) proposed the mathematical scheduling methods ; Afriat ( 1972 ) and Hanoch and Rothschild ( 1972 ) proposed ‘a assortment of trials of consistence of the observed informations with proficient and economic efficiency ‘ ( ibid ) . These methods all had their advantages when gauging frontier but it was Charnes, Cooper and Rhodes ( 1978 ) foremost introduced the informations enclosure analysis method, which was widely accepted and used when analyzing house or bank efficiency.

As discussed before in literature reappraisal, DEA is a non-parametric method for ciphering comparative efficiency tonss in a multi input-output production environment. It measures the public presentation of all decision-making units ( DMU ) compared to the generated efficient frontier. Best-practice Bankss that construct the DEA frontier, bring forth given end product combinations with the lowest degree of inputs or accomplish the highest degree of end product with a given degree of inputs. The efficiency mark for every DMU is ranged between 0 and 1 and those scored 1 DMU is the efficient DMU with regard to the remainder of the DMUs in the sample set. Compared to the parametric method ( stochastic frontier attack SFA ) , DEA do n’t necessitate a specified functional signifier for the cost or production map. Furthermore, DEA would non let any random mistake. SFA requires the specification of a cost map and detailed informations on input monetary values. Meanwhile, a really alone characteristic of Nipponese banking since the 1997 fiscal crisis was that ‘the diminution in employment in Japan will non ever have been commensurate with the diminution in banking activity ‘ ( Drake et al 2009, p.7 ) . Although the informations of labor disbursals are available for most Nipponese Bankss, informations of the figure of employees are barely found. The deficiency of the input informations would compromise the SFA consequence while DEA is widely accepted as an attack that could easy pattern slacks. Furthermore, DEA merely need small construction on efficient frontier and the ‘existence of an input/output correspondence ‘ ( Drake and Hall 2003, p.898 ) while disregarding the conditional premise that all houses had to confront the same production engineering.

Drake and Hall ( 2003 ) considered the Farrell ‘s theory is the most suited one for the individual output/two input instance. They illustrated the theory in the unit isoquant diagram ( Figure 2 ) below ( p.898 ) .

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Figure 2 Farrell efficiency

In Figure 2, the unit isoquant ( yy ) represents the possible combinations of the two inputs ( x1, x2 ) . These inputs are used to bring forth certain units of the individual end product ( Y ) . Point E represents the best pick of input mix as the cost is most minimized during the production procedure with regard to the given input monetary values which is the incline of WW ‘ . Any house or bank get to this point is considered as overall efficient. Meanwhile, point Q is an unfavorable pick as the input mix is inappropriate. Firm at the point is allocatively inefficient. Furthermore, the alteration of return to scale would be unsure if the house continued to utilize more input at this point Q. Point R represents the worst pick of input mix. For the same end product y, this point R needs much more input ten. It is so considered both allocatively inefficient and technically inefficient. The overall efficiency is the combination of allocative efficiency and proficient efficiency. For the ground of Nipponese banking informations job which mentioned before, the allocative efficiency would non be possible to discourse. Hence, this paper will concentrate on the proficient efficiency, which could be decomposed into scale efficiency and pure proficient efficiency because of uncertainness of the return to graduated table. It so could be concluded that DEA is the better method to measure Nipponese banking efficiency.

## Changeless Return to Scale ( CRS )

DEA method could break up to two chief theoretical accounts: changeless returns to scale theoretical account ( CRS ) and variable return graduated table theoretical account ( VRS ) .

CRS focuses on the long-term optimum degree of end products, on which it considers the best patterns were runing. Charnes, Cooper and Rhodes ( 1978 ) ab initio proposed the CCR theoretical account which is the basic DEA theoretical account to measure the efficiency. It is ‘built on the premise of CRS of activities as depicted for the production frontier in the individual input-single end product instance ‘ ( Cooper et al. 2006, p.83 ) . The cardinal thought of this theoretical account is to look into the ability of the house to bring forth a given degree of end products with most minimized usage of input. Cooper et Al. ( 2006 ) explained this basic theoretical account in a brief manner ( p.20-24 ) .

For each DMU, they formed the practical input and end product by weights ( ) and ( )

( i=1, 2, 3…s )

. ( r=1, 2, 3…m )

Linear scheduling is so used to maximise the ratio:

That is:

( 1.1 )

Subject to

( 1.2 )

( 1.3 )

( 1.4 )

In which, n is the figure of Bankss ; m is the figure of end products ; s is the figure of inputs ; o is a specific DMU ( bank ) to be evaluated ; is the sum of end product s from bank J ; is the sum of input m to bank J ; is weight chosen for end product s ; is weight chosen for input m. The look ( 1.1 ) indicates that each DMU maximizes its ain efficiency ratio by minimising its inputs to bring forth the same degree of end products. The restraint ( 1.2 ) prevent that the efficiency ratios of other DMUs transcending one ; restraint ( 1.3 ) and ( 1.4 ) indicate that all inputs and end products have nonzero worth and they make certain all the weights are positive.

Under the old premise of and tens are all positive, in ( 1.1 ) showed positive consequence for every J. Therefore, the consequence of multiply both sides of ( 1.2 ) with would remain positive. Because the fractional figure stay unchanged when multiplying of both numerator and denominator with the same figure, when set and presume the optimum consequences of additive plan are ( and ) , the fractional plan ( FPo ) could be replaced by a additive plan ( LPo ) as showed below:

( 1.5 )

Capable to ( 1.6 )

( 1.7 )

( J = 1, … , N )

( 1.8 )

( 1.9 )

Assume the optimum solutions to the additive plan are ( where constrained by ( 1.8 ) and ( 1.9 ) . If, it could be concluded that DMU is efficient in CCR theoretical account and there is at least one optimum ( ) . If, DMU is inefficient in CCR theoretical account.

Furthermore, an tantamount enclosure signifier is needed and could be derived from the additive plan by utilizing dichotomy in additive scheduling.

Subject to

represents the ( input-oriented ) efficiency mark of DMUo.

## Variable Tax returns to Scale ( VRS )

BBC theoretical account as an extension of CCR theoretical account was proposed by Banker et Al. ( 1984 ) . In this theoretical account, the production frontier was no longer a individual consecutive line. The bing DMUs extended frontier to a bulging joint-line. The alteration ( Figure 3 cited from Cooper et Al. 2006, p.84 ) raised the thought of variable returns-to-scale and those frontiers that showed piecewise additive characteristic. From Figure 3, Cooper et Al. ( 2006 ) concluded that ‘increasing returns-to-scale occurring in the first solid line section followed by diminishing returns-to-scale in the 2nd section and CRS happening at the point where the passage from the first to the 2nd section is made ‘ ( ibid ) .

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Figure 3 Cooper et Al. 2006

The tantamount BBC fractional plan could be written as follow:

Subject to

In which, and are vectors, is scalar and it could be any figure.

Similarly to CCR theoretical account, this fractional plan could so transform to the double multiplier signifier of additive plan. It is written as follow:

Subject to

Furthermore, the envelopment signifier of input-oriented BBC theoretical account could be written as follows:

Subject to

( j=1,2, … , N )

In which, ; ; vitamin E is a row vector with all elements integrity ; is a column vector with all elements greater or equal to zero.

If the consequence, the DMU here is efficient and represented by a frontier point. If, the DMU so could be concluded as inefficient. The nonzero optimal is the benchmark for a specific DMU. The mention set provides coefficients to specify the conjectural efficient DMU. It besides shows that in order to do the DMU efficient, it non merely indicates certain inputs should be decreased but besides points out the measure.

In add-on, the VRS focuses on short-term conditions compared to CRS looks frontward to the long-run. Both of the CCR and BBC theoretical account could be really utile when measuring proficient efficiency ( pure proficient efficiency and scale efficiency ) . Since scale efficiency is comparative to CRS and VRS, it is clearer when uniting CRS and VRS in Figure 3 into one figure ( Figure 4 ) .

proficient efficiency

Figure 4 Scale efficiency

I-O CRS TE =

I-O VRS TE =

I-O Scale Efficiency= = AB/AS

## The Malmquist Index

The Malmquist Index was theoretically introduced by Cave et Al. ( 1982 ) . It presents ‘the ration between two distance maps that compares a house ‘s productiveness with that of an alternate house and, in a straightforward dynamic extension, over clip ‘ ( Zofio 2007, p.2371 ) . F & A ; auml ; re et Al. ( 1992 ) proved that after break uping Malmquist Index, it could be implemented through DEA techniques. The decomposition consequence in two factors that might act upon steadfast under the inefficient production status: technological alteration and efficiency alteration. F & A ; auml ; re et Al. ( 1992 ) gave two footings for these two factors: catch-up term for technological alteration and frontier-shift term for efficiency alteration. The Catch-up term ( TEC ) measures the grade to which a DMU improves or worsens its efficiency between the two clip periods while Frontier-shift term ( FC ) measures the alteration in the efficient frontiers between the two clip periods.

Compared to original DEA methods, which evaluate comparative efficiency of DMUs at the really same clip, the DEA based Malmquist Index method is able to mensurate productiveness alteration over clip. In item, the consequences of this Malmquist biased DEA method are indexes for the entire factor productiveness growing of certain determination doing unit between two periods of clip. It so could be used for analyzing productiveness growing of houses before and after large event such as amalgamation or fiscal crisis.

Two individual period and two assorted period steps are needed when conculating Malmquist index ( MI ) . After presuming there were efficiency steps for clip period T and period t+1 and puting period T as the base twelvemonth, the Malmquist index so could be written as follow:

In which, is the comparative efficiency of a peculiar DMU in period t+1 compared to the period T frontier ; is the comparative efficiency of a peculiar DMU in period T compared to the period T frontier ; is the comparative efficiency of a peculiar DMU in period t+1 compared to the period t+1 frontier ; is the comparative efficiency of a peculiar DMU in period T compared to the period t+1 frontier. F & A ; auml ; re et Al. ( 1992 ) defined that if, the productiveness growing is positive, there was certain productiveness addition ; if, the productiveness growing is negative, there was certain productivity loss ; if, the productiveness growing remained, there was no alteration productiveness from clip T to t+1.

As discussed above, the Malmquist index could be decomposed to two constituents: TEC and FC. Hence, MI = TEC – FC, the illustration as follow:

MIgrapgh

Figure 5 Malmquist Productivity Index

## ;

Both TEC and FC portion same characteristic with the Malmquist Index as they were positive related. When, there were advancement in comparative efficiency and the frontier engineering from period T to t+1 ; when, there were reasoning backward in comparative efficiency and the frontier engineering from period T to t+1 ; if, there were no alteration in comparative efficiency and the frontier engineering from period T to t+1

In all, when sing the background and current state of affairs of Nipponese banking industry, the better method to analyse the efficiency of big commercial Bankss in Japan is Data Envelopment Analysis. DEA is capable of managing multiple inputs and end products. Furthermore, it has the advantage of leting inexplicitly specified functional signifier for the production map. As discussed above, the DEA based on Malmquist Index method is the suited one to measure efficiency of big Bankss in Japan before and after recent banking crisis.